Universal enveloping algebra

For the universal enveloping W* algebra of a C* algebra, see Sherman–Takeda theorem.

In mathematics, a universal enveloping algebra is the most general (unital, associative) algebra that contains all representations of a Lie algebra.

Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand-Naimark theorem, to contain the C* algebra of the corresponding Lie group. This relationship generalizes to the idea of Tannaka-Krein duality between compact topological groups and their representations.

From an analytic viewpoint, the universal enveloping algebra of the Lie algebra of a Lie group may be identified with the algebra of left-invariant differential operators on the group.

Informal construction

The idea of the universal enveloping algebra is to embed a Lie algebra ${\mathfrak {g}}$ into an associative algebra ${\mathcal {A}}$ with identity in such a way that the abstract bracket operation in ${\mathfrak {g}}$ corresponds to the commutator $xy-yx$ in ${\mathcal {A}}$. There may be many ways to make such an embedding, but there is one "largest" such ${\mathcal {A}}$, called the universal enveloping algebra of ${\mathfrak {g}}$.

Generators and relations

Let ${\mathfrak {g}}$ be a Lie algebra, assumed finite-dimensional for simplicity, with basis $X_{1},\ldots X_{n}$. Let $c_{ijk}$ be the structure constants for this basis, so that

$[X_{i},X_{j}]=\sum _{k=1}^{n}c_{ijk}X_{k}$.

Then the universal enveloping algebra is the associative algebra with identity generated by elements $x_{1},\ldots x_{n}$ subject to the relations

$x_{i}x_{j}-x_{j}x_{i}=\sum _{k=1}^{n}c_{ijk}x_{k}$

and no other relations.

Consider, for example, the Lie algebra sl(2,C), spanned by the matrices

$$X={\begin{pmatrix}0&1\\0&0\end{pmatrix}}\qquad Y={\begin{pmatrix}0&0\\1&0\end{pmatrix}}\qquad H={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}~,$$

which satisfy the commutation relations $[H,X]=2X$, $[H,Y]=-2Y$, and $[X,Y]=H$. The universal enveloping algebra of sl(2,C) is then the algebra generated by three elements $x,y,h$ subject to the relations

$hx-xh=2x,\quad hy-yh=-2y,\quad xy-yx=h$,

and no other relations. We cannot take the universal enveloping algebra to be the algebra of $2\times 2$ matrices (or a subalgebra thereof), because, for example, the matrix $X$ satisfies the additional relation $X^{2}=0$, which is not forced on us by the three defining relations of the universal enveloping algebra. That is to say, the product in the universal enveloping algebra is not the matrix product but a formal product in which only the three defining relations above are imposed. It turns out (as a consequence of the Poincaré–Birkhoff–Witt theorem) that the elements $1,x,x^{2},\dots$ are all linearly independent in the universal enveloping algebra. This is a universal property of universal enveloping algebras.

Finding a basis

In general, elements of the universal enveloping algebra are linear combinations of products of the generators in all possible orders. Using the defining relations of the universal enveloping algebra, we can always re-order those products in a particular order, say with all the factors of $x_{1}$ first, then factors of $x_{2}$, etc. For example, whenever we have a term that contains $x_{2}x_{1}$ (in the "wrong" order), we can use the relations to rewrite this as $x_{1}x_{2}$ plus a linear combination of the $x_{j}$'s. Doing this sort of thing repeatedly eventually converts any element into a linear combination of terms in the desired order. Thus, elements of the form

$x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{n}^{k_{n}}$

with the $k_{j}$'s being non-negative integers, span the enveloping algebra. (We allow $k_{j}=0$, meaning that we allow terms in which no factors of $x_{j}$ occur.) The Poincaré–Birkhoff–Witt theorem, discussed below, asserts that these elements are linearly independent and thus form a basis for the universal enveloping algebra. In particular, the universal enveloping algebra is always infinite dimensional.

The Poincaré–Birkhoff–Witt theorem implies, in particular, that the elements $x_{1},\ldots x_{n}$ themselves are linearly independent. It is therefore common—if potentially confusing—to identify the $x_{j}$'s with the generators $X_{j}$ of the original Lie algebra. That is to say, we identify the original Lie algebra as the subspace of its universal enveloping algebra spanned by the generators. It should be emphasized, however, that if ${\mathfrak {g}}$ is an algebra of $n\times n$ matrices, the universal enveloping of ${\mathfrak {g}}$ is not contained in the algebra of $n\times n$ matrices, since the universal enveloping algebra is always infinite dimensional. Thus, in the case of sl(2,C), if we identify our Lie algebra as a subspace of its universal enveloping algebra, we must now interpret $X$, $Y$ and $H$ not as $2\times 2$ matrices, but rather as elements of some abstract algebra.

Formalities

The formal construction of the universal enveloping algebra makes precise the idea of "no other relations." Specifically, we first take the tensor algebra of ${\mathfrak {g}}$ and then quotient it by the smallest two-sided ideal containing elements of the form $x_{i}x_{j}-x_{j}x_{i}-\sum c_{ijk}x_{k}$. The universal enveloping algebra is the quotient of the tensor algebra on generators subject to relations imposed by the structure constants; it is the most general unital associative algebra with a compatible Lie bracket with the original Lie algebra.

Formal definition

Recall that every Lie algebra ${\mathfrak {g}}$ is in particular a vector space. Thus, one is free to construct the tensor algebra $T({\mathfrak {g}})$ from it. The tensor algebra is a free algebra: it simply contains all possible tensor products of all possible vectors in ${\mathfrak {g}}$, without any restrictions whatsoever on those products.

That is, one constructs the space

$T({\mathfrak {g}})=K\,\oplus \,{\mathfrak {g}}\,\oplus \,({\mathfrak {g}}\otimes {\mathfrak {g}})\,\oplus \,({\mathfrak {g}}\otimes {\mathfrak {g}}\otimes {\mathfrak {g}})\,\oplus \,\cdots$

where $\otimes$ is the tensor product, and $\oplus$ is the direct sum of vector spaces. Here, K is the field over which the Lie algebra is defined. From here, through to the remainder of this article, the tensor product is always explicitly shown. Many authors omit it, since, with practice, its location can usually be inferred from context. Here, a very explicit approach is adopted, to minimize any possible confusion about the meanings of expressions.

The universal enveloping algebra is obtained by taking the quotient by imposing the relations

$a\otimes b-b\otimes a=[a,b]$

for all a and b in the embedding of ${\mathfrak {g}}$ in $T({\mathfrak {g}}).$ To avoid the tautological feeling of this equation, keep in mind that the bracket on the right hand side of this equation is actually the abstract "bracket" operation on the Lie algebra. Recall that the bracket operation on a Lie algebra is any bilinear map of ${\mathfrak {g}}\times {\mathfrak {g}}$ to ${\mathfrak {g}}$ that is skew-symmetric and satisfies the Jacobi identity. This bracket is not necessarily computed as $[X,Y]=XY-YX$ for some associative product structure on ${\mathfrak {g}}$. The goal of the universal enveloping algebra is to embed (in a canonical way) a Lie algebra into an associative algebra in such a way the abstract bracket operation on the original Lie algebra is now the commutator $ab-ba$ in that associative algebra.

To be more precise, the universal enveloping algebra is defined as the quotient space

$U({\mathfrak {g}})=T({\mathfrak {g}})/I$

where I is the two-sided ideal over $T({\mathfrak {g}})$ generated by elements of the form

$a\otimes b-b\otimes a-[a,b]$

Note that the above is an element of

${\mathfrak {g}}\oplus ({\mathfrak {g}}\otimes {\mathfrak {g}})\subset T({\mathfrak {g}})$

and so can be validly used to construct the ideal within $T({\mathfrak {g}})$. Thus, for example, given $a,b,c,d,f,g\in {\mathfrak {g}}$, one can write

$c\otimes d\otimes \cdots \otimes (a\otimes b-b\otimes a-[a,b])\otimes f\otimes g\cdots$

as an element of I, and all elements of I are obtained as linear combinations of elements of the above form. Clearly, $I\subset T({\mathfrak {g}})$ is a subspace. In essence, the universal enveloping algebra is what remains of the tensor algebra after modding out the Poisson algebra structure.

Superalgebras

The analogous construction for Lie superalgebras is straightforward; one need only to keep careful track of the sign, when permuting elements. In this case, the (anti-)commutator of the superalgebra lifts to an (anti-)commuting Poisson bracket.

One can obtain a different result by taking the above construction, and replacing every occurrence of the tensor product by the exterior product. That is, one uses this construction to create the exterior algebra of the Lie group; this construction results in the Gerstenhaber algebra, with the grading naturally coming from the grading on the exterior algebra. (This should not be confused with the Poisson superalgebra).

Other generalizations

The construction has also been generalized for Malcev algebras, Bol algebras and left alternative algebras.

Universal property

The universal enveloping algebra, or rather the universal enveloping algebra together with the canonical map $h:{\mathfrak {g}}\to U({\mathfrak {g}})$, possesses a universal property. Suppose we have any Lie algebra map

$\phi :{\mathfrak {g}}\to A$

to a unital associative algebra A (with Lie bracket in A given by the commutator). More explicitly, this means that we assume

$\phi ([X,Y])=\phi (X)\phi (Y)-\phi (Y)\phi (X)$

for all $X,Y\in {\mathfrak {g}}$. Then there exists a unique unital algebra homomorphism

${\widehat {\phi }}:U({\mathfrak {g}})\to A$

such that

$\phi ={\widehat {\phi }}\circ h$

where $h:{\mathfrak {g}}\to U({\mathfrak {g}})$ is the canonical map. (The map $h$ is obtained by embedding ${\mathfrak {g}}$ into its tensor algebra and then composing with the quotient map to the universal enveloping algebra. This map is an embedding, by the Poincare-Birkhoff-Witt theorem.)

To put it differently, if $\phi :{\mathfrak {g}}\rightarrow A$ is a linear map into a unital algebra $A$ satisfying $\phi ([X,Y])=\phi (X)\phi (Y)-\phi (Y)\phi (X)$, then $\phi$ extends to an algebra homomorphism of ${\widehat {\phi }}:U({\mathfrak {g}})\to A$. Since $U({\mathfrak {g}})$ is generated by elements of ${\mathfrak {g}}$, the map ${\widehat {\phi }}$ must be uniquely determined by the requirement that

${\widehat {\phi }}(X_{i_{1}}\cdots X_{i_{N}})=\phi (X_{i_{1}})\cdots \phi (X_{i_{N}}),\quad X_{i_{j}}\in {\mathfrak {g}}$.

The point is that because there are no other relations in the universal enveloping algebra besides those coming from the commutation relations of ${\mathfrak {g}}$, the map ${\widehat {\phi }}$ is well defined, independent of how one writes a given element $x\in U({\mathfrak {g}})$ as a linear combination of products of Lie algebra elements.

The universal property of the enveloping algebra immediately implies that every representation of ${\mathfrak {g}}$ acting on a vector space $V$ extends uniquely to a representation of $U({\mathfrak {g}})$. (Take $A=\mathrm {End} (V)$.) This observation is important because it allows (as discussed below) the Casimir elements to act on $V$. These operators (from the center of $U({\mathfrak {g}})$) act as scalars and provide important information about the representations. The quadratic Casimir element is of particular importance in this regard.

The tensor algebra on a vector space is the free functor from the category of vector spaces Vect to the category of algebras Alg which is left-adjoint to the forgetful functor mapping each algebra to its underlying vector space and each algebra homomorphism to its underlying linear map. The unit of this adjunction is the natural transformation of including each vector space V as the rank-one tensor product of itself in its tensor algebra T(V); the counit is the unique algebra homomorphism from the free algebra T(Y) on the underlying vector space of the algebra Y to Y given by evaluation of products and sums of elements of Y according to Y's rules of multiplication.

Let T be the functor defined as the composition of the tensor algebra functor on vector spaces composed with the forgetful functor of underlying vector spaces of Lie algebras.

This universal property of universal enveloping algebras follows from the tensor algebra as a natural transformation. That is, there is a functor T from the category of Lie algebras over K to the category of unital associative K-algebras, taking a Lie algebra to the corresponding free algebra. Similarly, there is also a functor U that takes the same category of Lie algebras to the same category of unital associative K-algebras. The two are related by a natural map that takes T into U: that natural map is the action of quotienting. The universal property passes through the natural map.

If A is any unital associative algebra, it naturally generates a Lie algebra A_L by taking the Lie bracket to be the commutator on A. This is a functor Lie from the category of algebras Alg to the category of Lie algebras LieAlg over some underlying field—in fact, it is a free functor. The functor U of universal enveloping algebras is left adjoint to the functor Lie, which maps an algebra A to the Lie algebra A_L. The two are adjoint, but certainly are not inverses: if we start with an associative algebra A, then U(A_L) is not equal to A; it is in general much bigger. (If, however, A is a commutative algebra, A_L is a trivial Lie algebra and its universal enveloping algebra will degenerate to A.) The unit of the adjunction is a natural embedding of A into A_L. The counit of the adjunction is the quotient of the universal enveloping algebra of a Lie algebra g with commutator Lie bracket by any other rules of the Lie algebra, for example that X² = 0 in sl(2, C) above. By functor composition, the universal enveloping algebra constructs an adjunction between the free Lie algebra on a vector space and the forgetful functor from Lie algebras to vector spaces.

Other algebras

Although the canonical construction, given above, can be applied to other algebras, the result, in general, does not have the universal property. Thus, for example, when the construction is applied to Jordan algebras, the resulting enveloping algebra contains the special Jordan algebras, but not the exceptional ones: that is, it does not envelope the Albert algebras. Likewise, the Poincaré–Birkhoff–Witt theorem, below, constructs a basis for an enveloping algebra; it just won't be universal. Similar remarks hold for the Lie superalgebras.

Poincaré–Birkhoff–Witt theorem

Main article: Poincaré–Birkhoff–Witt theorem

The Poincaré–Birkhoff–Witt theorem gives a precise description of $U({\mathfrak {g}})$. This can be done in either one of two different ways: either by reference to an explicit vector basis on the Lie algebra, or in a coordinate-free fashion.

Using basis elements

One way is to suppose that the Lie algebra can be given a totally ordered basis, that is, it is the free vector space of a totally ordered set. Recall that a free vector space is defined as the space of all finite supported functions from a set X to the field K (finitely supported means that only finitely many values are non-zero); it can be given a basis $e_{a}:X\to K$ such that $e_{a}(b)=\delta _{ab}$ is the indicator function for $a,b\in X$. Let $h:{\mathfrak {g}}\to T({\mathfrak {g}})$ be the injection into the tensor algebra; this is used to give the tensor algebra a basis as well. This is done by lifting: given some arbitrary sequence of $e_{a}$, one defines the extension of $h$ to be

$h(e_{a}\otimes e_{b}\otimes \cdots \otimes e_{c})=h(e_{a})\otimes h(e_{b})\otimes \cdots \otimes h(e_{c})$

The Poincaré–Birkhoff–Witt theorem then states that one can obtain a basis for $U({\mathfrak {g}})$ from the above, by enforcing the total order of X onto the algebra. That is, $U({\mathfrak {g}})$ has a basis

$e_{a}\otimes e_{b}\otimes \cdots \otimes e_{c}$

where $a\leq b\leq \cdots \leq c$, the ordering being that of total order on the set X. The proof of the theorem involves noting that, if one starts with out-of-order basis elements, these can always be swapped by using the commutator (together with the structure constants). The hard part of the proof is establishing that the final result is unique and independent of the order in which the swaps were performed.

This basis should be easily recognized as the basis of a symmetric algebra. That is, the underlying vector spaces of $U({\mathfrak {g}})$ and the symmetric algebra are isomorphic, and it is the PBW theorem that shows that this is so. See, however, the section on the algebra of symbols, below, for a more precise statement of the nature of the isomorphism.

Coordinate-free

One can also state the theorem in a coordinate-free fashion, avoiding the use of total orders and basis elements. This is convenient when there are difficulties in defining the basis vectors, as there can be for infinite-dimensional Lie algebras. It also gives a more natural form that is more easily extended to other kinds of algebras. This is accomplished by constructing a filtration $U_{m}{\mathfrak {g}}$ whose limit is the universal enveloping algebra $U({\mathfrak {g}}).$

First, a notation is needed for an ascending sequence of subspaces of the tensor algebra. Let

$T_{m}{\mathfrak {g}}=K\oplus {\mathfrak {g}}\oplus T^{2}{\mathfrak {g}}\oplus \cdots \oplus T^{m}{\mathfrak {g}}$

where

$T^{m}{\mathfrak {g}}=T^{\otimes m}{\mathfrak {g}}={\mathfrak {g}}\otimes \cdots \otimes {\mathfrak {g}}$

is the m-times tensor product of ${\mathfrak {g}}.$ The $T_{m}{\mathfrak {g}}$ form a filtration:

$K\subset {\mathfrak {g}}\subset T_{2}{\mathfrak {g}}\subset \cdots \subset T_{m}{\mathfrak {g}}\subset \cdots$

More precisely, this is a filtered algebra, since the filtration preserves the algebraic properties of the subspaces. Note that the limit of this filtration is the tensor algebra $T({\mathfrak {g}}).$

It was already established, above, that quotienting by the ideal is a natural transformation that takes one from $T({\mathfrak {g}})$ to $U({\mathfrak {g}}).$ This also works naturally on the subspaces, and so one obtains a filtration $U_{m}{\mathfrak {g}}$ whose limit is the universal enveloping algebra $U({\mathfrak {g}}).$

Next, define the space

$G_{m}{\mathfrak {g}}=U_{m}{\mathfrak {g}}/U_{m-1}{\mathfrak {g}}$

This is the space $U_{m}{\mathfrak {g}}$ modulo all of the subspaces $U_{n}{\mathfrak {g}}$ of strictly smaller filtration degree. Note that $G_{m}{\mathfrak {g}}$ is not at all the same as the leading term $U^{m}{\mathfrak {g}}$ of the filtration, as one might naively surmise. It is not constructed through a set subtraction mechanism associated with the filtration.

Quotienting $U_{m}{\mathfrak {g}}$ by $U_{m-1}{\mathfrak {g}}$ has the effect of setting all Lie commutators defined in $U_{m}{\mathfrak {g}}$ to zero. One can see this by observing that the commutator of a pair of elements whose products lie in $U_{m}{\mathfrak {g}}$ actually gives an element in $U_{m-1}{\mathfrak {g}}$. This is perhaps not immediately obvious: to get this result, one must repeatedly apply the commutation relations, and turn the crank. The essence of the Poincaré–Birkhoff–Witt theorem is that it is always possible to do this, and that the result is unique.

Since commutators of elements whose products are defined in $U_{m}{\mathfrak {g}}$ lie in $U_{m-1}{\mathfrak {g}}$, the quotienting that defines $G_{m}{\mathfrak {g}}$ has the effect of setting all commutators to zero. What PBW states is that the commutator of elements in $G_{m}{\mathfrak {g}}$ is necessarily zero. What is left are the elements that are not expressible as commutators.

In this way, one is lead immediately to the symmetric algebra. This is the algebra where all commutators vanish. It can be defined as a filtration $S_{m}{\mathfrak {g}}$ of symmetric tensor products ${\mbox{Sym}}^{m}{\mathfrak {g}}$. Its limit is the symmetric algebra $S({\mathfrak {g}})$. It is constructed by appeal to the same notion of naturality as before. One starts with the same tensor algebra, and just uses a different ideal, the ideal that makes all elements commute:

$S({\mathfrak {g}})=T({\mathfrak {g}})/(a\otimes b-b\otimes a)$

Thus, one can view the Poincaré–Birkhoff–Witt theorem as stating that $G({\mathfrak {g}})$ is isomorphic to the symmetric algebra $S({\mathfrak {g}})$, both as a vector space and as a commutative algebra.

The $G_{m}{\mathfrak {g}}$ also form a filtered algebra; its limit is $G({\mathfrak {g}}).$ This is the associated graded algebra of the filtration.

The construction above, due to its use of quotienting, implies that the limit of $G({\mathfrak {g}})$ is isomorphic to $U({\mathfrak {g}}).$ In more general settings, with loosened conditions, one finds that $S({\mathfrak {g}})\to G({\mathfrak {g}})$ is a projection, and one then gets PBW-type theorems for the associated graded algebra of a filtered algebra. To emphasize this, the notation ${\mbox{gr}}U({\mathfrak {g}})$ is sometimes used for $G({\mathfrak {g}}),$ serving to remind that it is the filtered algebra.

Other algebras

The theorem, applied to Jordan algebras, yields the exterior algebra, rather than the symmetric algebra. In essence, the construction zeros out the anti-commutators. The resulting algebra is an enveloping algebra, but is not universal. As mentioned above, it fails to envelop the exceptional Jordan algebras.

Left-invariant differential operators

Suppose $G$ is a real Lie group with Lie algebra ${\mathfrak {g}}$. Following the modern approach, we may identify ${\mathfrak {g}}$ with the space of left-invariant vector fields (i.e., first-order left-invariant differential operators). Specifically, if we initially think of ${\mathfrak {g}}$ as the tangent space to $G$ at the identity, then each vector in ${\mathfrak {g}}$ has a unique left-invariant extension. We then identify the vector in the tangent space with the associated left-invariant vector field. Now, the commutator (as differential operators) of two left-invariant vector fields is again a vector field and again left-invariant. We can then define the bracket operation on ${\mathfrak {g}}$ as the commutator on the associated left-invariant vector fields. This definition agrees with any other standard definition of the bracket structure on the Lie algebra of a Lie group.

We may then consider left-invariant differential operators of arbitrary order. Every such operator $A$ can be expressed (non-uniquely) as a linear combination of products of left-invariant vector fields. The collection of all left-invariant differential operators on $G$ forms an algebra, denoted $D(G)$. It can be shown that $D(G)$ is isomorphic to the universal enveloping algebra $U({\mathfrak {g}})$.

In the case that ${\mathfrak {g}}$ arises as the Lie algebra of a real Lie group, one can use left-invariant differential operators to give an analytic proof of the Poincaré–Birkhoff–Witt theorem. Specifically, the algebra $D(G)$ of left-invariant differential operators is generated by elements (the left-invariant vector fields) that satisfy the commutation relations of ${\mathfrak {g}}$. Thus, by the universal property of the enveloping algebra, $D(G)$ is a quotient of $U({\mathfrak {g}})$. Thus, if the PBW basis elements are linearly independent in $D(G)$—which one can establish analytically—they must certainly be linearly independent in $U({\mathfrak {g}})$. (And, at this point, the isomorphism of $D(G)$ with $U({\mathfrak {g}})$ is apparent.)

Algebra of symbols

The isomorphism of $U({\mathfrak {g}})$ and $S({\mathfrak {g}})$, as associative algebras, leads to the concept of the algebra of symbols $\star ({\mathfrak {g}})$. This is the space of symmetric polynomials, endowed with a product, the $\star$, that places the algebraic structure of the Lie algebra onto what is otherwise a standard associative algebra. That is, what the PBW theorem obscures (the commutation relations) the algebra of symbols restores into the spotlight.

The algebra is obtained by taking elements of $S({\mathfrak {g}})$ and replacing each generator $e_{i}$ by an indeterminate, commuting variable $t_{i}$ to obtain the space of symmetric polynomials $K[t_{i}]$ over the field $K$. Indeed, the correspondence is trivial: one simply substitutes the symbol $t_{i}$ for $e_{i}$. The resulting polynomial is called the symbol of the corresponding element of $S({\mathfrak {g}})$. The inverse map is

$w:\star ({\mathfrak {g}})\to U({\mathfrak {g}})$

that replaces each symbol $t_{i}$ by $e_{i}$. The algebraic structure is obtained by requiring that the product $\star$ act as an isomorphism, that is, so that

$w(p\star q)=w(p)\otimes w(q)$

for polynomials $p,q\in \star ({\mathfrak {g}}).$

The primary issue with this construction is that $w(p)\otimes w(q)$ is not trivially, inherently a member of $U({\mathfrak {g}})$, as written, and that one must first perform a tedious reshuffling of the basis elements (applying the structure constants as needed) to obtain an element of $U({\mathfrak {g}})$ in the properly ordered basis. An explicit expression for this product can be given: this is the Berezin formula. It follows essentially from the Baker–Campbell–Hausdorff formula for the product of two elements of a Lie group.

A closed form expression is given by

$p(t)\star q(t)=\left.\exp \left(t_{i}m^{i}\left({\frac {\partial }{\partial u}},{\frac {\partial }{\partial v}}\right)\right)p(u)q(v)\right\vert _{u=v=t}$

where

$m(A,B)=\log \left(e^{A}e^{B}\right)-A-B$

and $m^{i}$ is just $m$ in the chosen basis.

The universal enveloping algebra of the Heisenberg algebra is the Weyl algebra (modulo the relation that the center be the unit); here, the $\star$ product is called the Moyal product.

Representation theory

The universal enveloping algebra preserves the representation theory: the representations of ${\mathfrak {g}}$ correspond in a one-to-one manner to the modules over $U({\mathfrak {g}})$. In more abstract terms, the abelian category of all representations of ${\mathfrak {g}}$ is isomorphic to the abelian category of all left modules over $U({\mathfrak {g}})$.

The representation theory of semisimple Lie algebras rests on the observation that there is an isomorphism, known as the Kronecker product:

$U({\mathfrak {g}}_{1}\oplus {\mathfrak {g}}_{2})\cong U({\mathfrak {g}}_{1})\otimes U({\mathfrak {g}}_{2})$

for Lie algebras ${\mathfrak {g}}_{1},{\mathfrak {g}}_{2}$. The isomorphism follows from a lifting of the embedding

$i({\mathfrak {g}}_{1}\oplus {\mathfrak {g}}_{2})=i_{1}({\mathfrak {g}}_{1})\otimes 1\oplus 1\otimes i_{2}({\mathfrak {g}}_{2})$

where

$i:{\mathfrak {g}}\to U({\mathfrak {g}})$

is just the canonical embedding (with subscripts, respectively for algebras one and two). It is straightforward to verify that this embedding lifts, given the prescription above. See, however, the discussion of the bialgebra structure in the article on tensor algebras for a review of some of the finer points of doing so: in particular, the shuffle product employed there corresponds to the Wigner-Racah coefficients, i.e. the 6j and 9j-symbols, etc.

Also important is that the universal enveloping algebra of a free Lie algebra is isomorphic to the free associative algebra.

Construction of representations typically proceeds by building the Verma modules of the highest weights.

In a typical context where ${\mathfrak {g}}$ is acting by infinitesimal transformations, the elements of $U({\mathfrak {g}})$ act like differential operators, of all orders. (See, for example, the realization of the universal enveloping algebra as left-invariant differential operators on the associated group, as discussed above.)

Casimir operators

The center of $U({\mathfrak {g}})$ is $Z(U({\mathfrak {g}}))$ and can be identified with the centralizer of ${\mathfrak {g}}$ in $U({\mathfrak {g}})$. That is, since the elements of ${\mathfrak {g}}$ generate $U({\mathfrak {g}})$, any element of $U({\mathfrak {g}})$ that commutes with each Lie algebra element is in the center of $U({\mathfrak {g}})$. Thus, the center is directly useful for classifying representations of ${\mathfrak {g}}$.

For a finite-dimensional semisimple Lie algebra, the Casimir operators form a distinguished basis from the center $Z(U({\mathfrak {g}}))$. These may be constructed as follows.

From the PBW theorem, it is clear that all central elements are linear combinations of symmetric homogenous polynomials in the basis elements $e_{a}$ of the Lie algebra. The Casimir invariants are the irreducible homogenous polynomials of a given, fixed degree. That is, given a basis $e_{a}$, a Casimir operator of order $m$ has the form

$C_{(m)}=\kappa ^{ab\cdots c}e_{a}\otimes e_{b}\otimes \cdots \otimes e_{c}$

where there are $m$ terms in the tensor product, and $\kappa ^{ab\cdots c}$ is a completely symmetric tensor of order $m$ belonging to the adjoint representation. That is, $\kappa ^{ab\cdots c}$ can be (should be) thought of as an element of $\left({\mbox{ad}}_{\mathfrak {g}}\right)^{\otimes m}.$ Recall that the adjoint representation is given directly by the structure constants, and so an explicit indexed form of the above equations can be given, in terms of the Lie algebra basis; this is originally a theorem of Israel Gel'fand. That is, from $[x,C_{(m)}]=0$, it follows that

$f_{ij}^{\;\;k}\kappa ^{jl\cdots m}+f_{ij}^{\;\;l}\kappa ^{kj\cdots m}+\cdots +f_{ij}^{\;\;m}\kappa ^{kl\cdots j}=0$

where the structure constants are

$[e_{i},e_{j}]=f_{ij}^{\;\;k}e_{k}$

As an example, the quadratic Casimir operator is

$C_{(2)}=\kappa ^{ij}e_{i}\otimes e_{j}$

where $\kappa ^{ij}$ is the inverse matrix of the Killing form $\kappa _{ij}.$ That the Casimir operator $C_{(2)}$ belongs to the center $Z(U({\mathfrak {g}}))$ follows from the fact that the Killing form is invariant under the adjoint action.

The center of the universal enveloping algebra of a simple Lie algebra is given in detail by the Harish-Chandra isomorphism.

Rank

The number of algebraically independent Casimir operators of a finite-dimensional semisimple Lie algebra is equal to the rank of that algebra, i.e. is equal to the rank of the Cartan-Weyl basis. This may be seen as follows. For a d-dimensional vector space V, recall that the determinant is the completely antisymmetric tensor on $V^{\otimes d}$. Given a matrix M, one may write the characteristic polynomial of M as

$\det(tI-M)=\sum _{n=0}^{d}p_{n}t^{n}$

For a d-dimensional Lie algebra, that is, an algebra whose adjoint representation is d-dimensional, the linear operator

${\mbox{ad}}:{\mathfrak {g}}\to {\mbox{End}}({\mathfrak {g}})$

implies that ${\mbox{ad}}_{x}$ is a d-dimensional endomorphism, and so one has the characteristic equation

$\det(tI-{\mbox{ad}}_{x})=\sum _{n=0}^{d}p_{n}(x)t^{n}$

for elements $x\in {\mathfrak {g}}.$ The non-zero roots of this characteristic polynomial (that are roots for all x) form the root system of the algebra. In general, there are only r such roots; this is the rank of the algebra. This implies that the highest value of n for which the $p_{n}(x)$ is non-vanishing is r.

The $p_{n}(x)$ are homogeneous polynomials of degree d-n. This can be seen in several ways: Given a constant $k\in K$, ad is linear, so that ${\mbox{ad}}_{kx}=k\,{\mbox{ad}}_{x}.$ By plugging and chugging in the above, one obtains that

$p_{n}(kx)=k^{d-n}p_{n}(x).$

By linearity, if one expands in the basis,

$x=\sum _{i=1}^{d}x_{i}e_{i}$

then the polynomial has the form

$p_{n}(x)=x_{a}x_{b}\cdots x_{c}\kappa ^{ab\cdots c}$

that is, a $\kappa$ is a tensor of rank $m=d-n$. By linearity and the commutativity of addition, i.e. that ${\mbox{ad}}_{x+y}={\mbox{ad}}_{y+x},$, one concludes that this tensor must be completely symmetric. This tensor is exactly the Casimir invariant of order m.

The center $Z({\mathfrak {g}})$ corresponded to those elements $z\in Z({\mathfrak {g}})$ for which ${\mbox{ad}}_{x}(z)=0$ for all x; by the above, these clearly corresponds to the roots of the characteristic equation. One concludes that the roots form a space of rank r and that the Casimir invariants span this space. That is, the Casimir invariants generate the center $Z(U({\mathfrak {g}})).$

Example: Rotation group SO(3)

The rotation group SO(3) is of rank one, and thus has one Casimir operator. It is three-dimensional, and thus the Casimir operator must have order (3-1)=2 i.e. be quadratic. Of course, this is the Lie algebra of $A_{1}.$ As an elementary exercise, one can compute this directly. Changing notation to $e_{i}=L_{i},$ with $L_{i}$ belonging to the adjoint rep, a general algebra element is $xL_{1}+yL_{2}+zL_{3}$ and direct computation gives

$\det \left(xL_{1}+yL_{2}+zL_{3}-tI\right)=-t^{3}-(x^{2}+y^{2}+z^{2})t+2xyz$

The quadratic term can be read off as $\kappa ^{ij}=\delta ^{ij}$, and so the squared angular momentum operator for the rotation group is that Casimir operator. That is,

$C_{(2)}=L^{2}=e_{1}\otimes e_{1}+e_{2}\otimes e_{2}+e_{3}\otimes e_{3}$

and explicit computation shows that

$[L^{2},e_{k}]=0$

after making use of the structure constants

$[e_{i},e_{j}]=\epsilon _{ij}^{\;\;k}e_{k}$

Example: Pseudo-differential operators

A key observation during the construction of $U({\mathfrak {g}})$ above was that it was a differential algebra, by dint of the fact that any derivation on the Lie algebra can be lifted to $U({\mathfrak {g}})$. Thus, one is led to a ring of pseudo-differential operators, from which one can construct Casimir invariants.

If the Lie algebra ${\mathfrak {g}}$ acts on a space of linear operators, such as in Fredholm theory, then one can construct Casimir invariants on the corresponding space of operators. The quadratic Casimir operator corresponds to an elliptic operator.

If the Lie algebra acts on a differentiable manifold, then each Casimir operator corresponds to a higher-order differential on the cotangent manifold, the second-order differential being the most common and most important.

If the action of the algebra is isometric, as would be the case for Riemannian or pseudo-Riemannian manifolds endowed with a metric and the symmetry groups SO(N) and SO (P, Q), respectively, one can then contract upper and lower indices (with the metric tensor) to obtain more interesting structures. For the quadratic Casimir invariant, this is the Laplacian. Quartic Casimir operators allow one to square the stress–energy tensor, giving rise to the Yang-Mills action. The Coleman–Mandula theorem restricts the form that these can take, when one considers ordinary Lie algebras. However, the Lie superalgebras are able to evade the premises of the Coleman–Mandula theorem, and can be used to mix together space and internal symmetries.

Examples in particular cases

If ${\mathfrak {g}}={\mathfrak {sl}}_{2}$, then it has a basis of matrices

$h={\begin{pmatrix}-1&0\\0&1\end{pmatrix}},{\text{ }}g={\begin{pmatrix}0&1\\0&0\end{pmatrix}},{\text{ }}f={\begin{pmatrix}0&0\\1&0\end{pmatrix}}$

which satisfy the following identities under the standard bracket:

$[h,g]=-2g$, $[h,f]=-2f$, and $[g,f]=-h$

this shows us that the universal enveloping algebra has the presentation

$U({\mathfrak {sl}}_{2})={\frac {\mathbb {C} \langle x,y,z\rangle }{(xy-yx+2y,xz-zx+2z,yz-zy+x)}}$

as a non-commutative ring.

If ${\mathfrak {g}}$ is abelian (that is, the bracket is always 0), then $U({\mathfrak {g}})$ is commutative; and if a basis of the vector space ${\mathfrak {g}}$ has been chosen, then $U({\mathfrak {g}})$ can be identified with the polynomial algebra over K, with one variable per basis element.

If ${\mathfrak {g}}$ is the Lie algebra corresponding to the Lie group G, then $U({\mathfrak {g}})$ can be identified with the algebra of left-invariant differential operators (of all orders) on G; with ${\mathfrak {g}}$ lying inside it as the left-invariant vector fields as first-order differential operators.

To relate the above two cases: if ${\mathfrak {g}}$ is a vector space V as abelian Lie algebra, the left-invariant differential operators are the constant coefficient operators, which are indeed a polynomial algebra in the partial derivatives of first order.

The center $Z({\mathfrak {g}})$ consists of the left- and right- invariant differential operators; this, in the case of G not commutative, is often not generated by first-order operators (see for example Casimir operator of a semi-simple Lie algebra).

Another characterization in Lie group theory is of $U({\mathfrak {g}})$ as the convolution algebra of distributions supported only at the identity element e of G.

The algebra of differential operators in n variables with polynomial coefficients may be obtained starting with the Lie algebra of the Heisenberg group. See Weyl algebra for this; one must take a quotient, so that the central elements of the Lie algebra act as prescribed scalars.

The universal enveloping algebra of a finite-dimensional Lie algebra is a filtered quadratic algebra.

Hopf algebras and quantum groups

The construction of the group algebra for a given group is in many ways analogous to constructing the universal enveloping algebra for a given Lie algebra. Both constructions are universal and translate representation theory into module theory. Furthermore, both group algebras and universal enveloping algebras carry natural comultiplications that turn them into Hopf algebras. This is made precise in the article on the tensor algebra: the tensor algebra has a Hopf algebra structure on it, and because the Lie bracket is consistent with (obeys the consistency conditions for) that Hopf structure, it is inherited by the universal enveloping algebra.

Given a Lie group G, one can construct the vector space C(G) of continuous complex-valued functions on G, and turn it into a C*-algebra. This algebra has a natural Hopf algebra structure: given two functions $\phi ,\psi \in C(G)$, one defines multiplication as

$(\nabla (\phi ,\psi ))(x)=\phi (x)\psi (x)$

and comultiplication as

$(\Delta (\phi ))(x\otimes y)=\phi (xy),$

the counit as

$\epsilon (\phi )=\phi (e)$

and the antipode as

$(S(\phi ))(x)=\phi (x^{-1}).$

Now, the Gelfand-Naimark theorem essentially states that every commutative Hopf algebra is isomorphic to the Hopf algebra of continuous functions on some compact topological group G—the theory of compact topological groups and the theory of commutative Hopf algebras are the same. For Lie groups, this implies that C(G) is isomorphically dual to $U({\mathfrak {g}})$; more precisely, it is isomorphic to a subspace of the dual space $U^{*}({\mathfrak {g}}).$

These ideas can then be extended to the non-commutative case. One starts by defining the quasi-triangular Hopf algebras, and then performing what is called a quantum deformation to obtain the quantum universal enveloping algebra, or quantum group, for short.

See also

• Milnor–Moore theorem
• Harish-Chandra homomorphism

References

• Dixmier, Jacques (1996) [1974], Enveloping algebras, Graduate Studies in Mathematics, 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740
• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666
• Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, Providence, R.I.: American Mathematical Society, doi:10.1090/gsm/034, ISBN 978-0-8218-2848-9, MR 1834454
• Musson, Ian M. (2012), Lie Superalgebras and Enveloping Algebras, Graduate Studies in Mathematics, 131, Providence, R.I.: American Mathematical Society, ISBN 0-8218-6867-5, Zbl 1255.17001
• Shlomo Sternberg (2004), Lie algebras, Harvard University.
• Universal enveloping algebra in nLab